Stable Beliefs and Conditional Probability Spaces
Konstantinos Gkikas
Abstract:
This thesis aims to provide a conceptual framework that unifies Leitgeb’s theory of stable beliefs, Battigalli-Siniscalchi’s notion of strong belief and the notions of core, a priori, abnormal and conditional belief studied by Van Fraassen and Arló-Costa.
We will first present the difficulties of modeling qualitative notions of belief and belief revision in a quantitative probabilistic setting. On one hand we have the probability 1 proposal for belief, which seems to be materially wrong and on the other hand we have the Lockean thesis (or any version of it) which deprives us of the logical closure of belief (Lottery Paradox). We will argue that Leitgeb’s theory of stability of belief is a path between this Scylla and Charybdis.
The first goal of this thesis is to provide an extension of Leitgeb’s theory into non-classical probability settings, where we can condition on events with measure 0. We will define the notion of r-stable sets in Van Fraassen’s setting, using two-place functions to take conditional probability as primitive. We will then use the notion of r-stability to provide a definition of conditional belief similar to Leitgeb’s.
The second goal of this thesis is to develop a formal language that will express the notion of conditional beliefs. To do so, we will first define the structures called conditional probabilistic frames that will give us the semantics for the logic of conditional beliefs (rCBL) that we will present.
Finally, we will use the operators □(safe belief) and C(certainty) to lay the foundations of developing a logic of stable beliefs.